Published in Transactions of the American Mathematical Society, Volume 350, Issue 7, July 1, 1998, pages 2939-2951.
This article was first published in Transactions of the American Mathematical Society , published by the American Mathematical Society. Copyright © 1998 American Mathematical Society. The definitive version is available at http://dx.doi.org/10.1090/S0002-9947-98-01969-2.
Probabilistic proofs and interpretations are given for the q-binomial theorem, q-binomial series, two of Euler's fundamental partition identities, and for q-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.